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Use a CAS double-integral evaluator to find the integrals. Then reverse the order of integration and evaluate, again with a CAS.$$\int_{0}^{2} \int_{y^{3}}^{4 \sqrt{2 y}}\left(x^{2} y-x y^{2}\right) d x d y$$

$x^{2}=y^{\wedge} 3$,$x=4 \operatorname{sqrt}(2 y)$

Calculus 3

Chapter 15

Multiple Integrals

Section 2

Double Integrals over General Regions

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Lectures

04:18

In mathematics, an integral assigns numbers to functions in a way that can describe displacement, area, volume, and other concepts that arise by combining infinitesimal data. Integration is one of the two main operations of calculus, with its inverse operation, differentiation, being the other. The area above the x-axis adds to the total.

26:18

In mathematics, a double integral is an integral where the integrand is a function of two variables, and the integral is taken over some region in the Euclidean plane.

01:00

Use a CAS double-integral …

03:00

03:17

03:32

the given double Integral is 0 to 2 y cube to four under route to weigh X Square, way minus X Y Square on DX and Dy Way, and we have to estimate the value of this integral by using the graphing calculator. So use a graphing calculator on the value that were obtained from our graphing calculator as equals to 97.43 Now we'll have to write this integral by reversing the order of Integration. And then again, we have to find the value of this in trickle by reversing by using the graphing calculator. So, first of all, I'll be writing the in trickle by reversing the order of integration. For that, we know that these other limits for X that means X is it cools to y cube exit vehicles to four under route to weigh on. These are the limits for why that wins. Why is it calls to zero? By Isaac World Stood to. Now I'll be graphing these equations or graph paper to sketch the regional federation. So this is the Y axis on this It takes six is on this golf represents X is equal to four under, coached away on this red cough represents X equals two y cube on the head It is why is equal to two on dirt hair. Why is equal to zero on the reason off? Integration is this region. Now we have to write this a trickle by reversing the order of integration. So really it double integral X squared y minus X y square. Divide the X. Now we have to write the limits. Foreign Drago for that very dick or war Diggle strip. So here, why is equal to we know the equation of this line is X is equal to four on approached away. Or we can write X square is equals to 16 times to why or weaken right vice vehicles two X squared, divided but hard to do so the lower limit will be X squared divided but hard to do on the upper limit would be this and the equation of the skull is vice equals toe. White cube is equal to X from hair of why comes out to be as X to the power one divide by three Should upper limiters x to the power one by three. Now the limits for X would very between this line on this line. So here, X 00 on here. Excess eight. So you get it. So this is the integral area or print. By reversing the order of integration Now again, we'll be using our graphing calculator toe estimate the value of the sin trickle So it's come out to be as and 97.43 So finally we can conclude that the double integral 0 to 2 White cube four route to weigh X squared y minus X y square. The ex divi is equal to the double integral zero toe eight X squared, divided by 32 x to the power one by three x squared y minus X y square Do I d. X is equal to 97.43

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